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```					Frequency-Domain

Wu, Yihong
EE 491D
5-12-2005
Overview
n   Convergence Properties
n   Choice of Block Size
n   Methods
n   Computational Complexity
n   Providing a possible solution to the
computational complexity problem

n   Attaining a more uniform convergence
rate by exploiting the orthogonality
properties of DFT
n   BAF Diagram

n   BLMS Equations

n   Convergence behavior
BAF Diagram

BLMS Equations
§ From the conventional LMS, we could get the
following equations for BLMS
BLMS Equations(cont.)
n       Tap-weight updated once after collection of every
block of data samples

n L is the block length
n     is the effective step-size parameter
nError is the difference between the output
and the desired signal
BLMS Equations(cont.)
n   If written in matrices format:
Convergence Properties
n   Average time constant equation:

n   If    is the same for both conventional LMS and BLMS, they
would have the same average time constant.

n   For zero-order formula, the      will be small while comparing
to 1/    .

n   For fast adaptation,        is small and L is large, so   will be
so large that the higher order effects would cause instability
problems. BLMS could overcome the problem.
Choice of Block Size
n   L = M: optimal choice for computational
complexity

n   L < M: offers the advantage of reduced
processing delay

n   L > M: gives rise to redundant operations in

n   Overall, L = M is the best choice
Methods
n   Overlap-Save Sectioning

n   Circular Convolution
Overlap-Save Sectioning
§   Linear convolution between a finite-length sequence
[w(n)]and an infinite-length sequence [x(n)].

§   Zero-padding w(n) from N-point to 2N-point

§   FFT both x(n) and w(n) to get Y(k)

§   First N point data would be ignored

§   Only looking for the last N point data
Overlap-Save Sectioning

Figure 2: Overlap-Save Sectioning.
Overlap-Save Sectioning

Figure 3: Overlap-Save FDAF
Overlap-Save Sectioning
n   FDAF Algorithm Based on
Overlap-Save Sectioning
n   Input signal is different

nOutput y(k)
nError E(k)

n   FDAF Algorithm Based on Overlap-Add
Sectioning
Circular Convolution
n   Reducing complexity
n   Constraint matrices have been
eliminated
n   Rank of F is M = N
u Data is not overlapping any more
u Error is always linear between Y(K)
and D(K)
Circular Convolution

Figure 5: Circular-Convolution FDAF.
Circular Convolution
n   FDAF Algorithm Based on Circular
Convolution
Computational Complexity
n   Equation

nDepending on the filter size
nGreatest: Linear convolution

nSmallest: Circular convulution
Computational Complexity

Table 1: FDAF computational
complexity ratios
Conclusion
n   Computational Complexity
n   Some methods
n   Potential to be developed in
the future
Thank you very much!

Have a nice Summer!

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